One-way quantum computation with two-photon multiqubit cluster states

One-way quantum computation with two-photon multiqubit cluster states Giuseppe Vallone,1, ∗ Enrico Pomarico,1, † Francesco De Martini,1, 2, ∗ and Paolo Mataloni1, ∗

arXiv:0807.3887v1 [quant-ph] 24 Jul 2008

1 Dipartimento di Fisica dell’Universit´ a Sapienza di Roma and Consorzio Nazionale Interuniversitario per le Scienze Fisiche della Materia, Roma, 00185 Italy 2 Accademia Nazionale dei Lincei, via della Lungara 10, Roma 00165, Italy (Dated: July 24, 2008)

We describe in detail the application of four qubit cluster states, built on the simultaneous entanglement of two photons in the degrees of freedom of polarization and linear momentum, for the realization of a complete set of basic one-way quantum computation operations. These consist of arbitrary single qubit rotations, either probabilistic or deterministic, and simple two qubit gates, such as a c-not gate for equatorial qubits and a universal c-phase (CZ) gate acting on arbitrary target qubits. Other basic computation operations, such as the Grover’s search and the Deutsch’s algorithms, have been realized by using these states. In all the cases we obtained a high value of the operation fidelities. These results demonstrate that cluster states of two photons entangled in many degrees of freedom are good candidates for the realization of more complex quantum computation operations based on a larger number of qubits. PACS numbers: 03.67.Mn, 03.67.Lx

I.

INTRODUCTION

The relevance of cluster states in quantum information and quantum computation (QC) has been emphasized in several papers in recent years [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. By these states novel significant tests of quantum nonlocality, which are more resistant to noise and show significantly larger deviations from classical bounds can be realized [4, 12, 13, 14]. Besides that, cluster states represent today the basic resource for the realization of a quantum computer operating in the one-way model [1]. In the standard QC approach any quantum algorithm can be realized by a sequence of single qubit rotations and two qubit gates, such as c-not and c-phase on the physical qubits [15, 16, 17], while deterministic one-way QC is based on the initial preparation of the physical qubits in a cluster state, followed by a temporally ordered pattern of single qubit measurements and feed-forward (FF) operations [1]. By exploiting the correlations existing between the physical qubits, the unitary gates on the so called “encoded” (or logical) qubit [5] are realized. In this way, nonunitary measurements on the physical qubits correspond to unitary gates on the logical qubits. It is precisely this non-unitarity of the physical process that causes the irreversibility nature (i.e. its “one-way” feature) of the model. By this model the difficulties of standard QC, related to the implementation of two qubit gates, are transferred to the preparation of the state. The FF operations, that depend on the outcomes of the already measured qubits and are necessary for a deterministic computation, can be classified in two classes:

∗ URL:

http://quantumoptics.phys.uniroma1.it/ ´ address: Univ. de Gen` eve GAP-Optique, Rue de l’Ecolede-M` edecine 20, CH-1211 Gen` eve 4, Suisse

† Present

i) the intermediate feed-forward measurements, i.e. the choice of the measurement basis; ii) the Pauli matrix feed-forward corrections on the final output state. The first experimental results of one-way QC, either probabilistic or deterministic, were demonstrated by using 4-photon cluster states generated via post-selection by spontaneous parametric down conversion (SPDC). [5, 8]. The detection rate in such experiments, approximately 1 Hz, was limited by the fact that four photon events in a standard SPDC process are rare. Moreover, four-photon cluster states are characterized by limited values of fidelity, while efficient computation requires the preparation of highly faithful states. More recently, by entangling two photons in more degrees of freedom, we created four-qubit cluster states at a much higher level of brightness and fidelity [14]. Precisely, this was demonstrated by entangling the polarization (π) and linear momentum (k) degrees of freedom of one of the two photons belonging to a hyperentangled state [18, 19]. Moreover, working with only two photons allows to reduce the problems related to the limited quantum efficiency of detectors. Because of these characteristics, two-photon four-qubit cluster states are suitable for the realization of high speed one-way QC [20, 21, 22]. In this paper we give a detailed description of the basic QC operations performed by using two-photon four-qubit cluster state, such as arbitrary single qubit rotations, the c-not gate for equatorial qubits and a c-phase gate. We verified also the equivalence existing between the two degrees of freedom for qubit rotations, by using either k or π as output qubit, demonstrating that all four qubits can be adopted for computational applications. Moreover, we also show the realization of two important basic algorithms by our setup, namely the Grover’s search algorithm and the Deutsch’s algorithm. The former identifies the item tagged by a “Black Box”, while the latter allows

2



the 4-qubit linear (or horseshoe) cluster




⊂ ⊃ |Φlin 4 i =|Φ4 i = |Φ4 i = 1 = (|+i1 |0i2 |0i3 |+i4 + |+i1 |0i2 |1i3 |−i4 2 + |−i1 |1i2 |0i3 |+i4 + |−i1 |1i2 |1i3 |−i4 ) ,

FIG. 1: (color online). Effect of measurements on a generic cluster state. a) Measurements in the computational basis {|0i, |1i} are indicated by red crosses. b) The cluster state after the measurements of some qubits in the computational basis.

to distinguish in a single run if a function is constant or balanced. The paper is organized as follows. In Sec. II we review the one-way model of QC realized through single qubit measurements on a cluster state. We also describe the basic building blocks that can be used in order to implement any general algorithm. In Sec. III we give a description of the source used to generate the two-photon four-qubit cluster state by manipulating a polarizationmomentum hyperentangled state. We describe in Sec. IV three basic operations realized by our setup: a generic single qubit rotation, a c-not gate for equatorial target qubit and a c-phase gate for fixed control and arbitrary target qubit. In Sec. V two explicit examples of quantum computation are given by the realization of the Grover’s search algorithm and the the Deutsch’s algorithm. Finally, the conclusions are given in Sec. VI.

II.

ONE-WAY COMPUTATION

Cluster states are quantum states associated to n dimensional lattices. It has been shown that twodimensional cluster states are a universal resource for QC [23]. The explicit expression of a cluster state is found by associating to each dot j of the lattice (see fig. 1) a qubit in the state |+ij = √12 (|0ij + |1ij ) and to each link between two adjacent qubits i and j, a c-phase gate CZij : CZij = |0ii h0| ⊗ 11j + |1ii h1| ⊗ σz(j) .

(1)

Considering a lattice L with N sites, the corresponding cluster state is then given by the expression:   Y  |ΦL (2) CZij  |+iN , Ni ≡ i,j linked

where |+iN ≡ |+i1 ⊗ |+i2 · · · ⊗ |+iN . Some explicit examples of cluster states are the 3-qubit linear cluster, 1 |Φlin 3 i = √ (|+i1 |0i2 |+i3 + |−i1 |1i2 |−i3 ) , 2

(3)

(4)

corresponding to four qubits linked in a row (see Fig. 4I)) and the 4-qubit Box cluster |Φ 4i=

1 (|0i1 |+i2 |0i3 |+i4 + |0i1 |−i2 |1i3 |−i4 2 + |1i1 |−i2 |0i3 |−i4 + |1i1 |+i2 |1i3 |+i4 ) . (5)

corresponding to four qubits linked in a square (see Fig. 10(left)). For a given cluster state, the measurement of a generic qubit j performed in the computational basis {|0ij , |1ij } (Fig. 1a)) simply corresponds to remove it and its relative links from the cluster (Fig. 1b)). In this way we obtain, up to possible σz corrections, a cluster state with N − 1 qubits: Y L\{j} (6) (σz(k) )sj |ΦN −1 i , |ΦL Ni → k∈Nj

where sj = 0 if the measurement output is |0ij , while sj = 1 if the measurement output is |1ij . In the previous equation Nj stands for the set of all sites linked with the qubit j. Then, by starting from a large enough square lattice, it is possible to create any cluster state associated to smaller lattices. In the following figures we will indicate with a red cross the measurement of a physical qubit performed in the computational basis, as shown in Fig. 1. Let’s now explain how the computation proceeds. Each algorithm consists of a measurement pattern on a specific cluster state. This pattern has a precise temporal ordering. It is well known that one-way computation doesn’t operate directly on the physical qubits of the cluster state on which measurements are performed. The actual computation takes place on the so-called encoded qubits, written nonlocally in the cluster through the correlation between physical qubits. We will use i, j = 1, · · · , N for the physical qubits and a, b = 1, · · · , M for the encoded qubits (M < N ). Some physical qubits (precisely M ) represent the input qubits of the computation (all prepared in the state |+iE ) and the corresponding dots can be arranged at the left of the graph. We then measure N − M qubits, leaving M physical qubits unmeasured, hence the output of computation will correspond (up to Pauli errors) to the unmeasured qubits. It’s possible to arrange the position of the dots in such a way that the time ordering of the measurement pattern goes from left to right. The computation proceeds by the measurement performed in the basis Bj (ϕ) ≡ {|ϕ+ ij , |ϕ− ij } ,

(7)

3 √ where H is the Hadamard gate H = 1/ 2(σx + σz ) and Rz (α) = e−iασz /2 is a rotation around the z axis in the Bloch sphere. The σx operations depends on the measurement output (s1 ) of the first qubit. This operation can be understood by noting that by measuring the first qubit of the state CZ12 |χi1 |+i2 in the basis B1 (α) we will obtain |χ′ i2 . This simple algorithm can be repeated by using two CBB1 in a row. By measuring qubit 1 in the bases B1 (α) the encoded qubit then is transformed into |χ′ iE and the encoded qubit moves from left to right into the cluster. By measuring qubit 2 in the basis B2 (β) the encoded qubit is now written in qubit 3 as |χ′′ iE : |χ′′ iE ≡ σxs2 HRz (β)σxs1 HRz (α)|χiE . FIG. 2: (color online). Cluster building blocks (CBB). For each CBBi we indicate the measurement on the physical cluster (left) and the corresponding operation on the logical qubits (right).

where |ϕ± ij ≡ √12 (eiϕ/2 |0ij ± e−iϕ/2 |1ij ). Here sj = 0 or sj = 1 if the measurement outcome of qubit j is |ϕ+ ij or |ϕ− ij respectively. The specific choice of ϕ for every physical qubit is determined by the measurement pattern. Note that the choice of the measurement basis for a specific qubit can also depend on the outcome of the already measured qubits: these are what we call feedforward measurements (type i)). In general, active modulators (as for example Pockels cells in case of polarization qubit) are required to perform the FF measurements. In some case, however, when more than one qubit is encoded in the same particle through different degrees of freedom (DOF’s), it is possible to perform FF measurement without the use of active modulators. This will be discussed in Sec. IV, when the measurement basis of the generic qubit j, encoded in one particle, depends only on the outputs of some other qubits encoded in the same particle. One-way computation can be understood in terms of some basic operations, the so-called cluster building blocks (CBB) (see Fig. 2). By combining different CBB’s it becomes possible to perform computation of arbitrary complexity [24]. We introduce here a convenient notation: by writing explicitely a state |χi close to a dot j (see Fig. 2), we indicate that the total state could be equivalently obtained by preparing the qubit j in the state |χij before applying the necessary CZ gates. CBB1 : Qubit rotation. Consider two qubits linked together like CBB1 shown in Fig. 2. Here the first qubit is initially prepared in the state |χi and the second qubit is arbitrary linked with other dots. By measuring qubit 1 in the basis B1 (α) we remove it from the cluster but we transfer the information into qubit 2 leaving its links unaltered. This corresponds to the following operation on the encoded qubit |χiE |χiE → |χ′ iE ≡ σxs1 HRz (α)|χiE ,

(8)

(9)

In this case the computation can be understood by observing that with the measurement of qubits 1 and 2 of the state CZ12 CZ23 |χi1 |+i2 |+i3 in the basis B1 (α) and B2 (β), we obtain |χ′′ i3 . CBB2 : c-phase gate. Consider two qubits linked in a column. This block simply performs a c-phase gate (CZ) between the two qubits. CBB3 : c-phase gate+rotation. If we have 3 qubits in a column and we measure the second qubit in the basis B2 (π/2) we remove it from the cluster but the information is transferred to the qubit 1 and 3. Precisely, on the logical qubit, this measurement realizes a c-phase gate followed by two single qubit rotations Rz (−π/2) (see Fig. 2). By combining these CBB’s we can obtain any desired quantum algorithm, written in general as |ψout i = UΣ Ug

M Y

a=1

|+ia ,

(10)

where M are the number of logical qubits, Ug is the unitary gate that the algorithm must perform and UΣ are the so called Pauli errors corrections [24, 25]: UΣ =

M Y

(σx[a] )xa (σz[a] )za .

(11)

a=1

The numbers xa , za = 0, 1 depend on the outcomes of all the single (physical) qubit measurements and determine the FF corrections (type ii)) that must be realized, at the end of the measurement pattern, in order to achieve deterministic computation. We indicate by the symbol [a] σz that the Pauli matrix σz acts on the logical qubit a. Note that if the output of the algorithm is one among NM the 2M states of the computational basis, i.e. a=1 |ra ia (ra = 0, 1), only the σx ’s of the unitary UΣ act nontrivially by flipping some qubits. The Pauli errors are then reduced to UΣ′ =

M Y

(σx[a] )xa .

a=1

(12)

4  











 



 



  







    

 

FIG. 3: (color online). Source of two-photon four-qubit cluster state. The hyperentangled states |Ξ±± i derive from the simultaneous entanglement on the polarization and linear momentum degrees of freedom. Polarization entanglement is obtained by double passage of the pump and SPDC pair through a BBO type I crystal and a lambda/4 wave plate. Mode selection performed by a four hole screen allows linear momentum entanglement. The half wave plate HW transforms the hyperentangled state |Ξ+− i into the cluster state |C4 i.

In this way the “errors” can be simply corrected by relabeling the output, and there is no need of active feedforward corrections on the quantum state. If, by measurQ ing the output qubits, we get the result M |sa ia (sa = Qa=1 M 0 or sa = 1) we must interpret it as a=1 |sa ⊕ xa ia with the Pauli errors given by the equation (12). This relabeling operation can be performed for example by an external classical computer.

III.

EXPERIMENTAL SETUP

In our experiment we generated two-photon fourqubit cluster states by starting from polarization(π)momentum(k) hyperentangled photon pairs obtained by SPDC (see Fig. 3). The hyperentangled states |Ξ±± i ≡ |Φ± iπ ⊗ |ψ ± ik were generated by a β-barium-borate (BBO) type I crystal pumped in both sides by a cw Argon laser beam (λp = 364nm) (cfr. 3). The detailed explanation of hyperentangled state preparation was given in previous papers [18, 26], to which we refer for details. In the above expression of |Ξ±± i we used the Bell states 1 |Φ± iπ = √ (|HiA |HiB ± |V iA |V iB ) , 2 1 |ψ ± ik = √ (|ℓiA |riB ± |riA |ℓiB ), 2

(13)

where |Hi, |V i correspond to the horizontal (H) and vertical (V ) polarizations and |ℓi, |ri refer to the left (ℓ) or right (r) paths of the photon A (Alice) or B (Bob) (see Fig. 3). In |Ξ±± i, the first signs refer to the polarization state |φ± iπ and the second ones to the linear momentum state |ψ ± ik . Starting from the state |Ξ+− i = |Φ+ iπ ⊗ |ψ − ik and applying a c-phase (CZ) gate between the control (kA ) and the target (πA ) qubits of the photon A, we generated

Stabilizers −zA zB −xA xB ZA zA XA XB ZA ZB −yA yB ZA −zB XA XB −ZA ZB zA zB YA XB xA yB −ZB yA yB XA YB xA YB −YA XB yA xB −xA xB ZB −YA YB zA Y A Y B zB −XA YB yA xB 11

S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15 S16

Tr[ρexp Sk ] 0.9941 ± 0.0011 0.8486 ± 0.0031 0.9372 ± 0.0035 0.9105 ± 0.0024 0.8386 ± 0.0032 0.9354 ± 0.0035 0.8963 ± 0.0044 0.7455 ± 0.0042 0.8215 ± 0.0034 0.8139 ± 0.0037 0.7944 ± 0.0037 0.8498 ± 0.0031 0.9350 ± 0.0036 0.9346 ± 0.0037 0.8186 ± 0.0035 1

TABLE I: Expectation values of the stabilizer operators Si .

the cluster state 1 |C4 i = (|HℓiA |HriB − |HriA |HℓiB 2 + |V riA |V ℓiB + |V ℓiA |V riB ) 1 1 (14) = √ |Φ+ iπ |ℓiA |riA − √ |Φ− iπ |riA |ℓiA 2 2 1 1 = √ |HiA |HiA |ψ + ik + √ |V iA |V iA |ψ − ik . 2 2 In the experiment, the CZ gate is realized by a zero-order half wave (HW) plate inserted on the rA mode, and corresponds to introduce a π phase shift for the vertical polarization of the rA output mode. It is worth noting that, at variance with the four-photon cluster state generation, the state (14) is created without any kind of postselection[32]. By using the correspondence |Hi ↔ |0i, |V i ↔ |1i, |ℓi ↔ |0i, |ri ↔ |1i, the generated state |C4 i ⊂ ⊃  is equivalent to |Φlin 4 i, |Φ4 i, |Φ4 i or |Φ4 i up to single qubit unitaries: lin |C4 i = U1 ⊗ U2 ⊗ U3 ⊗ U4 |Φlin 4 i ≡ U|Φ4 i .

(15)

|Φlin 4 i

With and |C4 i we will refer to the cluster state expressed in the “cluster” and “laboratory” basis respectively. The explicit expression of the unitaries Uj depends on the specific ordering of the physical qubits (kA , kB , πA and πB ) and will be indicated in each case in the following. The basis changing will be necessary in order to know which are the correct measurements we need to perform in the actual experiment. In general if on qubit j the chosen algorithm requires a measurement in the basis |α± ij , the actual measurement basis in the laboratory is given by Uj |α± ij . In order to characterize the generated state we used the stabilizer operator formalism [27]. It can be shown [28] that 16

|C4 ihC4 | =

1 X Sk , 16 k=1

(16)

5 where the Sk are the so called stabilizer operators Sk |C4 i = |C4 i, ∀k = 1, · · · , 16 (see table I). The fidelity of the experimental cluster ρexp can be measured by 16

F|C4 i

1 X Tr[ρexp Sk ] , = Tr[ρexp |C4 ihC4 |] = 16

(17)

k=1

i.e. by measuring the expectation values of the stabilizer operators. In table I we report the stabilizer operators for |C4 i and the corresponding experimental expectation values. The obtained fidelity is F = 0.880±0.013, demonstrating the high purity level of the generated state. Cluster states were observed at 1 kHz detection rate. The two photons are detected by using interference filter with bandwidth ∆λ = 6nm. In table I we use the following notation for polarization Zj = |Hij hH| − |V ij hV | Yj = i|V ij hH| − i|Hij hV | Xj = |Hij hV | + |V ij hH|

j = A, B

(18)

j = A, B

(19)

and linear momentum operators zj = |ℓij hℓ| − |rij hr| yj = i|rij hℓ| − i|ℓij hr| xj = |ℓij hr| + |rij hℓ|

for either Alice (A) or Bob (B) photons.

FIG. 4: (color online). Measurement pattern for single qubit rotations. (a) Top: arbitrary single qubit rotations on a four qubit linear cluster state are carried out in three steps (I, II, III). In each measurement, indicated by a red cross, the information travels from left to right. Bottom: equivalent logical circuit.

II: Measuring qubit 2 in the basis B2 (α), the logical qubit (now encoded in qubit 3) is transformed into i s |χ′ i = (σx ) 2 HRz (α)|χin i, with Rz (α) = e− 2 ασz . III: Measurement of qubit 3 is performed in the basis:

IV.

BASIC OPERATIONS WITH 2-PHOTON CLUSTER STATE

In this section we describe the implementation of simple operations performed with the generated four-qubit two-photon cluster state. A.

Single qubit rotations

In the one-way model a three-qubit linear cluster state (simply obtained by the four-qubit cluster by measuring the first qubit) is sufficient to realize an arbitrary single qubit transformation[33] |χin i → Rx (β)Rz (α)|χin i, where Rz (α) = e−iασz /2 and Rx (β) = e−iβσx /2 . The algorithm consists of two CBB1 ’s on a row. Precisely, by using the four-qubit cluster expressed in the cluster basis the following measurement pattern must be followed (see Fig. 4): I: A three-qubit linear cluster is generated by measuring the first qubit in the computational basis {|0i1 , |1i1 }. As said, in Section II this operation removes the first qubit from the cluster and gener(2) ates (σz )s1 |Φlin 3 i. The input logical qubit |χin i is then encoded in qubit 2. If the outcome of the first measurement is |0i1 then |χin i = |+i, otherwise |χin i = |−i.

• B3 (β) if s2 = 0

• B3 (−β) if s2 = 1 This represents a FF measurement (type i)) since the choice of measurement basis depends on the previous outputs. This operation leaves the last qubit in the state |χout i = i s (σx ) 3 HRz [(−1)s2 β] |χ′ i with Rx (β) = e− 2 βσx . Then, by using some simple Pauli matrix algebra, the output state (encoded in qubit 4) can be written as |χout i = σxs3 σzs2 Rx (β)Rz (α)|χin i .

(20)

In this way, by suitable choosing α and β, we can perform any arbitrary single qubit rotation |χin i → Rx (β)Rz (α)|χin i up to Pauli errors (σxs3 σzs2 ), that should be corrected by proper feed-forward operations (type ii)) to achieve a deterministic computation[8]. In our case we applied this measurement pattern by considering different ordering of the physical qubits. In this way we encoded the output qubit either in the polarization or in the linear momentum of photon B, demonstrating the QC equivalence of the two degrees of freedom. The measurement apparatus is sketched in Fig. 5. The k modes corresponding to photons A or B, are respectively matched on the up and down side of a common symmetric beam splitter (BS) (see inset), which can

6 


      

 

  






 




 

















 







   




 
 




















 














FIG. 5: (color online). Measurement setup for photons A and B. Momentum qubits kA and kB are measured by two thin glass (φA , φB ), acting as phase shifters and inserted before a common 50:50 BS. Polarization qubits πA and πB are measured by standard tomographic setup (Dπ ). BS and Dπ outputs are indicated by sj = 0, 1, where the index j refers to the corresponding DOF. FF correction apparatus for deterministic QC [used in our experiment only with ordering (a)] is given by a 35 m length single mode fiber and two Pockels cells (σx , σz ) driven by the output signals of detector a1 , a3 , a4 . Dashed lines for H, BS and FF correction apparatus indicate that these devices can be inserted or not in the setup depending on the particular measurement (see text for details). Inset: tomographic apparatus Dπ and spatial mode matching on the BS.

πB

α 0 −π/2 −π/2 −π/2

β π/2 0 π/2 −π/4

F (s2 = 0, s3 = 0) 0.908 ± 0.006 0.942 ± 0.004 0.913 ± 0.005 0.899 ± 0.007

F (s2 = 0, s3 = 1) 0.860 ± 0.008 0.946 ± 0.004 0.929 ± 0.004 0.898 ± 0.006

πB

α 0 −π/2 −π/2 −π/2

β π/2 0 π/2 −π/4

F (s2 = 1, s3 = 0) 0.932 ± 0.005 0.873 ± 0.006 0.851 ± 0.007 0.928 ± 0.007

F (s2 = 1, s3 = 1) 0.935 ± 0.006 0.847 ± 0.006 0.848 ± 0.007 0.932 ± 0.007

TABLE II: (color online). Polarization (πB ) experimental fidelities (F) of single qubit rotation output states for different values of α and β. Each datum is obtained by the measurements of the different Stokes parameters, each one lasting 10 sec.

be also finely adjusted in the vertical direction such that one or both photons don’t pass through it. Polarization analysis is performed by a standard tomography apparatus Dπ (λ/4, λ/2 and polarizing beam splitter PBS). Depending of the specific measurement the HWs oriented at 22.5◦ are inserted to perform the Hadamard operation H in the apparatus Dπ . They are used together with the λ/4 in order to transform the {|ϕ+ iπA , |ϕ− iπA } states into linearly polarized states. Two thin glass plates before the BS allow to set the basis of the momentum measurement for each photon. Let’s consider the following ordering of the physical

FIG. 6: (color online). Polarization (πB ) output Bloch vectors of single qubit rotations. The experimental results (arrows) are shown with their projections on theoretical directions (dashed lines). Arrow colours correspond to different values of α and β (see table II).

qubits (see Eq. (15)): a)

(1,2,3,4)=(kB , kA , πA , πB ), U =σx H ⊗ σz ⊗ 11 ⊗ H .

(21)

The output state, encoded in the polarization of photon B, can be written in the laboratory basis as |χout iπB = σz sπA σx skA HRx (β)Rz (α)|χin i ,

(22)

where the H gate derives from the change between the cluster and laboratory basis. This also implies that the actual measurement bases are BkB (0) for the momentum of photon B (qubit 1) and BkA (α + π) (i.e. |α∓ ikA ) for the momentum of photon A (qubit 2). According to the one-way model, the measurement basis on the third qubit (πA ) depends on the results of the measurement on the second qubit (kA ). These are precisely what we call FF measurements (type i)). In our scheme this simply corresponds to measure πA in the bases BπA (β) or BπA (−β), depending on the BS output mode (i.e. skA = 0 or skA = 1). These deterministic FF measurements are a direct consequence of the possibility to encode two qubits (kA and πA ) in the same photon. As a consequence, at variance with the case of four-photon cluster states, in this case active feed-forward measurements (that can be realized by adopting Pockels cells) are not required, while Pauli errors corrections are in any case necessary for deterministic QC. We first realized the experiment without FF corrections (in this case we didn’t use the retardation fiber

7

FIG. 7: (color online). Momentum (kB ) output Bloch vectors of single qubit rotations. The experimental results (arrows) are shown with their projections on theoretical directions (dashed lines). Arrow colours correspond to different values of α and β (see table III).

kB

α(β = 0) 0 π/2 π/4 −π/4

F (s2 = 0, s3 = 0) 0.961 ± 0.003 0.879 ± 0.006 0.998 ± 0.005 0.833 ± 0.007

F (s2 = 0, s3 = 1) 0.971 ± 0.003 0.895 ± 0.005 0.961 ± 0.006 0.956 ± 0.006

kB

α(β = 0) 0 π/2 π/4 −π/4

F (s2 = 1, s3 = 0) 0.944 ± 0.0044 0.799 ± 0.007 0.919 ± 0.008 0.946 ± 0.008

F (s2 = 1, s3 = 1) 0.943 ± 0.005 0.918 ± 0.005 0.857 ± 0.009 0.872 ± 0.008

TABLE III: Momentum (kB ) experimental fidelities (F) of single qubit rotation output states for different values of α and β. Each datum is obtained by the measurements of the different Stokes parameters, each one lasting 10 sec.

and the Pockels cells in the setup). The results obtained for s2 = s3 = 0 (i.e. when the computation proceeds without errors) with |χin i = |+i are shown in fig. 6(a). We show on the Bloch sphere the experimental output qubits and their projections on the theoretical state HRx (β)Rz (α)|+i for the whole set of α and β. The corresponding fidelities are given in table II. We also performed the tomographic analysis (shown in fig 6b),c),d)) on the output qubit πB for all the possible combinations of s2 and s3 and for the input qubit |χin i = |±i. The high fidelities obtained in these measurements indicate that deterministic QC can be efficiently implemented in this configuration by Pauli errir active FF corrections.

FIG. 8: (color online). Output fidelities of the single qubit rotation algorithm with [FF, black (blue) columns] or without [NO FF, gray (orange) columns] feed forward. In both cases, the four columns of the histograms refer to the measurement of the output state (encoded in the polarization of photon B) by detector b1 in coincidence with a1 , . . . , a4 respectively. Grey dashed columns (THEO) correspond to theoretical fidelities in the no-FF case.

They were realized by using the entire measurement apparatus of fig. 5. Here two fast driven transverse LiN bO3 Pockels cells (σx and σz ) with risetime = 1n sec and V λ ∼ 1KV are activated by the output signals of 2 detectors ai (i = 1, 3, 4) corresponding to the different values of sπA and skA . They perform the operation σz sπA σx skA on photon B, coming from the output skB = 0 of BS and transmitted through a single mode optical fiber. Note that no correction is needed when photon A is detected on the output a2 (sπA = skA = 0). Temporal synchronization between the activation of the high voltage signal and the transmission of photon B through the Pockels cells is guaranteed by suitable choice of the delays D. We used only one BS output of photon B, namely skB = 0, in order to perform the algorithm with initial state |χin i = |+i. The other BS output corresponds to the algorithm starting with the initial state |χin i = |−i. By referring to Fig. 5, each detector aj corresponds to a different value of skA and sπA . Precisely, a1 corresponds to skA = 0 and sπA = 1 and activates the Pockels cell σz (see eq. 22). Detector a2 corresponds to skA = sπA = 0, i.e. the computation without errors and thus no Pockels cell is activated. Detector a3 corresponds to skA = 1, sπA = 0 and activates σx , while a4 corresponds to skA = sπA = 1 and both σx and σz are activated. In fig. 8 the output state fidelities obtained with/without active FF corrections (i.e. turning on/off the Pockels cells) are compared for different values of α and β. The expected theoretical fidelities in the no-FF case are also shown. In all the cases the computational errors are corrected by the FF action, with average mea-

8 O H

O 11

α Control output π/2 s1 = 0 → |1ic s1 = 1 → |0ic π/4 s1 = 0 → |1ic s1 = 1 → |0ic

F (s4 = 0) 0.965 ± 0.004 0.972 ± 0.004 0.995 ± 0.008 0.946 ± 0.010

F (s4 = 1) 0.975 ± 0.004 0.973 ± 0.004 0.902 ± 0.012 0.945 ± 0.009

α Control output F (s1 = s4 = 0) F (s1 = 0, s4 = 1) π/2 |0ic ≡ |ℓikB 0.932 ± 0.004 0.959 ± 0.003 |1ic = |rikB 0.941 ± 0.005 0.940 ± 0.005 π/4 |0ic = |ℓikB 0.919 ± 0.007 0.932 ± 0.007 |1ic = |rikB 0.878 ± 0.009 0.959 ± 0.006

TABLE IV: Experimental fidelity (F) of c-not gate output target qubit for different value of α and O.

FIG. 9: (color online). c-not and c-phase gates by the four qubit cluster. (top) c-not gate realization via measurement of qubits 1, 4 on the horseshoe cluster and equivalent circuit. (bottom) c-phase gate realization via measurement of qubits 1 and 2 in the bases B1 (α) and B2 (α) and equivalent circuit.

sured fidelity F = 0.867 ± 0.018. The overall repetition rate is about 500Hz, which is more than 2 orders of magnitude larger than one-way single qubit-rotation realized with 4-photon cluster states. We also demonstrated the computational equivalence of the two DOF’s of photon B by performing the same algorithm with the following qubit ordering: b)

(1,2,3,4)=(πB , πA , kA , kB ), U =H ⊗ σz ⊗ σx ⊗ σz H .

(23)

In this case we used the momentum of photon B (kB ) as output state. The explicit expression of the output state |χout ikB in the laboratory basis is now |χout ikB = s s (σz ) 3 (σx ) 2 σz HRx (β)Rz (α)|χin i . By using only detectors a2 , a3 , b1 , b2 in fig. 5 we measured |χout ikB for different values of α (which correspond in the laboratory to the polarization measurement bases |α∓ iπA ) and β = 0 (which correspond in the laboratory to the momentum bases | − β± ikA ). The first qubit (πB ) was always measured in the basis |±iπB . The kB tomographic analysis for all the possible values of s2 ≡ sπA and s3 ≡ skA are shown in Fig. 7. i.e. for different values of s1 ≡ sπB ). We obtained an average value of fidelity F > 0.9 (see table III). In this case the realization of deterministic QC by FF corrections could be realized by the adoption of active phase modulators. The (π)-(k) computational equivalence and the use of active feed-forward show that the multidegree of freedom approach is feasible for deterministic one-way QC. B.

c-not gate

The four qubit cluster allows the implementation of nontrivial two-qubit operations, such as the c-not gate.

Precisely, a c-not gate acting on a generic target qubit belonging to the equatorial plane of the Bloch sphere (i.e. a generic state of the form √12 (|0i + eiγ |1i)) can be realized by the four-qubit horseshoe (180◦ rotated) cluster state |Φlin 4 i (see eq. (4)) Let’s consider Fig. 9(top). The measurement of qubits 1 and 4 realizes a c-not gate (the logical circuit shown in figure) between the control |+ic and target |+it qubit. By measuring the qubit 1 in the basis {|0i1 , |1i1 } or |±i1 we realized on the control input qubit |+ic the gate O = 11 or O = H respectively. The measurement of qubit 4 in the basis |α± i4 realizes the gate HRz (α) on the target input qubit |+it . The algorithm is concluded by the vertical link that perform a CZ operation. The input state |+ic ⊗|+it is transformed, in case of no “errors” (i.e. s1 = s4 = 0), into |Ψout i = CZct (O|+ic ⊗ Ht Rz (α)|+it ) = Ht c-not(O|+ic ⊗ Rz (α)|+it ). This circuit realizes the c-not gate (up to the Hadamard Ht ) for arbitrary equa−iα/2 torial target qubit (since Rz (α)|+i = e √2 (|0i+eiα |1i)) and control qubit |0i, |1i or |±i depending on the measurement basis of qubit 1. The experimental realization of this gate is performed by adopting the following ordering between the physical qubits: c)

(1,2,3,4)=(kA , kB , πB , πA ), U =σz H ⊗ σx ⊗ 11 ⊗ H .

(24)

In this case the control output qubit is encoded in the momentum kB , while the target output is encoded in the polarization πB . In order to compensate the Ht gate arising from the cluster algorithm we inserted two Hadamard in the polarization analysis of the detectors b1 and b2 (see fig. 5). The output state in the laboratory basis is then written as |Ψout i = (Σ)s4 σx(c) c-not(Oσzs1 |+ic ⊗ Rz (α)|+it ) , (25) where all the possible measurement outcomes of qubits 1 (c) (t) and 4 are considered. The Pauli errors are Σ = σz σz , (c) while the matrix σx is due to the changing between cluster and laboratory basis.

9 α 0 π π/2 −π/2 π/2 π/2 π/4

β 0 0 0 0 π/2 −π/2 π/2

FkB (kA = |−i) 0.878 ± 0.004 0.919 ± 0.003 0.876 ± 0.005 0.880 ± 0.004 0.969 ± 0.002 0.950 ± 0.003 0.885 ± 0.006

FkB (kA = |+i) 0.933 ± 0.003 0.917 ± 0.004 0.816 ± 0.005 0.883 ± 0.004 0.949 ± 0.003 0.939 ± 0.003 0.916 ± 0.005

TABLE V: Experimental fidelity (F) of c-phase gate output target qubit for different value of α and β. In parenteses we indicate the corresponding measured output of the control qubit kA .

By measuring kA in the basis {|ℓikA , |rikA } we perform the O = H operation on the control qubit. By looking at eq. (25), this means that if skA ≡ s1 = 0 (skA = 1) the control qubit is |1i (|0i), while the target qubit is Rz (α)|+it (σx Rz (α)|+it ). In this case the gate acts on a control qubit |0i or |1i, without any superposition of these two states. We first verified that the gate acts correctly in this situation. In Table IV(top) we report the experimental fidelities (F ) of the output target qubit πB for two different values of α and for the two possible values of s4 . The high values of F show that the gate works correctly when the control qubit is |0i or |1i. The second step was to verify that the gate works correctly with the control qubit in a superposition of |0i and |1i. This was realized by measuring the qubit kA in the basis |±ikA and performing the O = 11 operation on the control qubit. The output state is written (without errors) as |Ψout i = |1ic ⊗ Rz (α)|+it + |0ic ⊗ σx Rz (α)|+it . In Table IV(bottom) we show the values of the experimental fidelities of the target qubit πB , corresponding to the measurement of the output control qubit kB in the basis {|0i, |1i} when O = 11. The results demonstrate the high quality of the operation also in this case.

C.

c-phase gate

The four qubit cluster allows also the realization of a c-phase gate for arbitrary target qubit and fixed control |+ic . The measurement pattern needed for this gate is shown in Fig. 9(bottom) and consists of the measurements of qubits 1 and 2 in the bases B1 (α) and B2 [(−1)s1 β]. These two measurements realize a generic rotation Rx (β)Rz (α) on the input target qubit |+it , as explained in subsection IV A. The link existing between qubit 3 and 4 in the cluster performs the subsequent cphase gate between the control qubit |+ic and a generic target qubit Rx (β)Rz (α)|+it , The experimental realization is done by considering the following ordering between the physical qubits: d)

(1,2,3,4)=(πA , πB , kB , kA ), U =H ⊗ 11 ⊗ σx ⊗ σz H.

(26)

FIG. 10: (color online). Grover algorithm. Left: realization of the Grover algorithm through the measurements of qubit 1 and 4 on the box cluster |Φ 4 i. Right: logical circuit implementing the Grover operator G = I · T . The first part of G is implemented by the cluster algorithm while the second part by the measurement.

We realized a c-phase gate for arbitrary target qubit and fixed control |+ic (see Fig. 9(c)) by measuring qubits 1 s1 and 2 of |Φlin 4 i in the bases |α± i and |(−) β± i respectively. By considering ordering d) we encoded the output state in the physical qubits kA and kB . For s1 = s2 = 0, by using the appropriate base changing, the output state is written as |Ψout i = |−ikA ⊗ σx |ΦikB + |+ikA ⊗ σx σz |ΦikB . (27) Here |ΦikB = Rx (β)Rz (α)|+i and the matrix σx is due to the basis changing. The measured fidelity of the target kB corresponding to a control |+ikA (|−ikA ) for different values of α and β are shown in table V. We obtaining an average value F = 0.907 ± 0.010 (F = 0.908 ± 0.011). V. A.

ALGORITHMS Grover algorithm

The Grover’s search algorithm for two input qubits is implemented by using the four qubit cluster state [5, 8, 29, 30]. Let’s describe the algorithm in general. Suppose to have 2M elements (encoded in M qubits) and a black box (or oracle) that tags one of them. The tagging, denoted as T , is realized by changing the sign of the desired element. The goal is to identify the tagged item by repeated query √ to the black box; the Grover’s algorithm requires O( 2M ) operations, while the best classical algorithm takes on average 2M /2 calculations. The general algorithm starts with the input state prepared as |Ψ+ i ≡ |+iE1 · · · |+iEM and consists of repeated applications of the Grover operator G, given by the oracle tagging T followed by the so called inversion about average operation I ≡ 2|Ψ+ ihΨ+ |−√ 11. We can thus write G ≡ I · T . In general after R = O( 2M ) iterations of G the tagged item is obtained at the output of the circuit with high probability. In the case of 2 qubits the quantum algorithm (shown in Fig. 10(right)) requires just one G operation. The four elements are |0iE1 |0iE2 , |0iE1 |1iE2 , |1iE1 |0iE2 and

10

e) (1,2,3,4)=(kB , πA , kA , πB ), U =σx H ⊗ H ⊗ σz H ⊗ H ,

(28)

the generated state (14) is equivalent to the box cluster |Φ 4 i up to the single qubit unitaries given by U. By this change of basis we can determine the correct measurement to be performed in the laboratory basis. The experimental results are shown in Fig. 11. In the upper graph we show the experimental fidelities obtained when the computation proceeds without Pauli errors, i.e. s1 = s4 = 0. The mean probability to identify the tagged item is 0.9482 ± 0.0080 and the algorithm is realized at ∼ 250Hz. We report in the lower graph the probability of identification when the FF corrections are implemented. The tagged item is discovered with probability 0.9475 ± 0.0022 and the algorithm is realized at ∼ 1kHz repetition rate, as expected. Note that, in the lower graph, a change of the tagged item corresponds to reorder the histograms. This is due to the fact that the measurement in the basis B(π) = {|−i, |+i} is the same as B(0) = {|+i, |−i}: the





















 





























 





|1iE1 |1iE2 . They are prepared in a complete superposition, i.e. in the state |+iE1 |+iE2 , while the black box tagging acts simply by changing the sign to one of the elements, for instance |1i|0i → −|1i|0i. It consists of a c-phase gate followed by two single qubit rotations, Rz (α)1 and Rz (β)2 . By setting the rotation angles αβ to 00, π0, 0π or ππ the black box tags respectively the states |1i|1i, |1i|0i, |0i, |1i, or |0i|0i (remember that Rz (π) is σz up to a global phase). The inversion operation consists of a c-phase gate and single qubit gates (see Fig. 10(right)). The inversion acts such as the output state of the system is exactly the tagged item. This algorithm can be realized in the one-way model by using the four-qubit box cluster previously defined. By measuring qubit 1 and 4 in the basis B1 (α) and B4 (β) we implement the black box and the first part of the inversion algorithm (Box cluster algorithm in Fig. 10). The output qubits are then encoded into the physical qubit 2 and 3. The Hσz operation needed to conclude the inversion operation can be performed at the measurement stage. Indeed we can measure the qubit 2 and 3 in the basis Bj (π): this is equivalent to apply Hσz gates and then perform the measurement in the computational basis {|0i, |1i} (see Measurement in Fig. 10). Without Pauli errors the desired tagged item is given by |s2 i|s3 i. Depending on the measurement outcome (s1 and s4 ) the corresponding Pauli errors are (σz )s1 (σx )s4 on the qubit E1 and (σz )s4 (σx )s1 on the qubit E2 . However, since the output of the algorithm will be one of the four states of computational basis, the σz operation leaves the output unchanged, while σx flips the output state (see equation(12)). In this way the tagged item is found to be |s2 ⊕ s4 iE1 |s3 ⊕ s1 iE2 and the FF corrections are simply relabeling FF. Let’s now describe the experimental realization of the Grover algorithm by our apparatus. If we consider the following ordering of the physical qubits (see equation (15)):

















 





























 

FIG. 11: (color online). Experimental results of the Grover algorithm. Upper graph: we report, for different tagged item, the probability of the different output states when the computation proceeds without Pauli errors. Experimental errors are of the order of 0.005 for higher histograms, while for the lower ones becomes 0.0005. Lower graph: experimental probabilities in the FF case. Experimental errors are of the order of 0.002 for higher histograms, while for the lower ones becomes 0.0004. Each datum is obtained by 10s measurement.

BB

Constant functions f1 f2 [e] 11a ⊗ 11e 11a ⊗ σx

Balanced functions f3 f4 [e] CNOTae (11a ⊗ σx )CNOTae

TABLE VI: Oracle operation (BB) depending on the single qubit function fi . The two function f1 and f2 are constant, while f3 and f4 are balanced.

difference is that in the first case we associate s = 0 to |−i while in the second we associate s = 0 to |+i. B.

Deutsch algorithm

The four qubit cluster state allow the implementation of the Deutsch’s algorithm for two input qubits [31]. This algorithm distinguishes two kinds of functions f (x) acting on a generic M -bit query input: the constant function returns the same value (0 or 1) for all input x and the balanced function gives 0 for half of the inputs and 1 for other half. Usually the function is implemented by a black box (or oracle). By the Deutsch’s algorithm one needs to query the oracle just once, while by using deterministic classical algorithms one needs to know the

11

Constant Function 



























































Balanced Function

FIG. 12: (color online). Implementation of the Deutsch algorithm. Measurement pattern on the physical cluster (left) and corresponding operations on the logical qubits (right). Measurements of qubit 2 and 4 on the left correspond to the Cluster algorithm in the right. The output logical qubits are encoded in the physical qubits 1 (query, q) and 3 (ancilla, e) that should be measured in order to read out the answer of the algorithm (Measurement in the right).

output of the oracle many times (as 2M−1 + 1). The oracle implements the function f on the query input |xiq through an ancillary qubit |yie : Oracle:|xiq |yie → |xiq |y ⊕ f (x)ie ,

(29)

where y = 0, 1 and x = 0, 1, · · · , 2M −1. If the oracle acts on the input qubits |+i1 |+i2 · · · |+iM |−ie , the output state is √

1 2M

M 2X −1

x=0

(−1)f (x) |xiq |−ie .

(30)

By applying Hadamard gates for each qubits the output state can be written as  M O    |0ia |1ie if f is constant   a=1 (31) M  O    |1ia |1ie if f is balanced  a=1

Then by measuring the query state in the computational basis we can discover if the function f is constant or balanced. The algorithm thus proceeds in three steps: • preparation: it consists in the initialization of the input state into |+i1 |+i2 · · · |+iM |−ie . • BB : this is the Oracle operation (29). • readout : apply Hadamard gates for each qubits and measure them in the computational basis {|0i, |1i}.





























































FIG. 13: (color online). Experimental probabilities of the output states of the Deutsch algorithm in case of constant (top) or balanced (bottom) function. The probabilities are shown for all the values of s4 and s2 , while in the last column are shown the results after the FF relabeling operation.

In the two-qubit version the function f acts on a single qubit |xiq . In this case there are 4 possible functions f on a single qubit: two are constant, namely f1 (0) = f1 (1) = 0 and f2 (0) = f2 (1) = 1, while two are balanced f3 (0) = 0, f3 (1) = 1 and f4 (0) = 1, f4 (1) = 0. Let’s describe the oracle operation (29) as a “Black Box” (BB). In Table VI we give the oracle operation on the two qubits |xiq |yie depending on the chosen function fi . By the four qubit cluster is possible to implement the two-qubit version of the algorithm. Let’s consider Fig. 12. The algorithm is implemented by the measurements of qubit 2 and 4, while the output is encoded in the qubit 1 (query qubit) and 3 (ancillary qubit). Precisely, the measurement of qubit 4 in the basis B4 (π) performs the transformation HRz (π) on the ancillary qubit. The BB is implemented by the measurement of qubit 2. If the oracle choose the measurement basis {|0i2 , |1i2 } it implements the f1 function. In fact the qubit 2 is removed from the cluster and no operations are performed on the other qubits. The full cluster algorithm consists in the operation 11q ⊗ [HRz (π)]e . This is exactly the Deutsch algorithm in the case of constant function up to an Hadamard gate on the query qubit to be implemented in the final measurement step. The measurement basis on the output qubits 1 and 3 are then B1 (0) and {|0i3 , |1i3 } (see Measurement in Fig. 12). By choosing B2 ( π2 ) as measurement basis for qubit 2 the oracle implements the f3 function and the obtained operation is (see CBB3 ) [Rz (− π2 )]q ⊗ [Rz (− π2 )]e ·CZqe . Together with the 11q ⊗ [HRz (π)]e the full cluster algo-

12 rithm becomes [Rz (− π2 )]q ⊗ [Rz (− π2 )H]e ·CNOTqe · 11q ⊗ [Rz (π)]e (the cluster algorithm in Fig. 12). This is the Deutsch algorithm in case of balanced function up to [Rz (− π2 )]q ⊗ [Rz (− π2 )H]e to be corrected in the final measurement step. These corrections corresponds to the choice of the measurement basis on the output qubits 1 and 3 as B1 (− π2 ) and {|0i3 , |1i3 } (see Measurement in Fig. 12). Without Pauli errors the output state of the Deutsch algorithm is given by |s1 = 0iq |s3 = 1ie in case of f1 and |s1 = 1iq |s3 = 1ie in case of f3 (see eq. (31)). The correct outcome considering the Pauli errors are in this case |s1 ⊕ s2 iq |s3 ⊕ s4 ie

|s1 ⊕ s2 ⊕ s4 iq |s3 ⊕ s4 ie

for f1 for f3

• the repetition rate of computation is almost three order of magnitude larger; • the fidelity of the computational operations is much higher (nearly 90%, even with active FF); • in some cases the intermediate FF operations do not require, differently from the case of multiphoton cluster states, active modulators. The FF operations are automatically (and also deterministically) implemented in these cases because of the entanglement existing between the two DOF’s of the same particle; • working with two photons allows to minimize the problems caused by the limited value of the detection efficiency.

(32)

Note that the BB operation obtained by the function f2 (f4 ) is essentially the same, up to a global phase, with respect to the function f1 (f3 ). In the following we then show only the results obtained in the case of f1 and f3 . In Fig. 13 we show the experimental probabilities of the different output states in function of the value of s2 and s4 for the f1 and f3 case. In the case of f1 , for s2 = s4 = 0 the output of the algorithm is the state |0iq |1ie with probability 0.924 ± 0.005, while in the other cases are |0 ⊕ s2 iq |1 ⊕ s4ie consistently with eq. (32). By using the FF relabeling we obtain the correct output with probability 0.949±0.002. In the case of balanced function the correct output after the FF operation is obtained with probability 0.967 ± 0.002. VI.

are the polarization and the linear momentum. Oneway QC based on multi-DOF cluster states presents some important advantages with respect to the one performed with multiphoton cluster states. In particular:

CONCLUSIONS

A larger number of qubits is necessary to perform more complex gates and algorithms. For instance, using a Type I NL crystal a continuum of k-emission modes is virtually available to create a multiqubit spatial entangled state. Even if the number of modes scales exponentially with the number of qubits, it is still possible to obtain a reasonable number (six or even eight) of qubits. Hence, increasing the number of DOF’s of the photon allows to move further than what expected by increasing the number of photons. Because of all these reasons we believe that cluster states based on many DOF’s may represent a good solution for experimental QC with photons on a mid-term perspective.

We have described the basic principles of operation of a one-way quantum computer operating with cluster states that are formed by two photons entangled in two different DOF’s. We have also presented the experiment (and the relative results) carried out when the DOF’s

This work was supported by Finanziamento Ateneo 06.

[1] R. Raussendorf and H. J. Briegel, Phys. Rev. Lett. 86, 5188 (2001). [2] M. A. Nielsen, Phys. Rev. Lett. 93, 040503 (2004). [3] D. E. Browne and T. Rudolph, Phys. Rev. Lett. 95, 010501 (2005). [4] N. Kiesel, C. Schmid, U. Weber, G. T´ oth, O. G¨ uhne, R. Ursin, and H. Weinfurter, Phys. Rev. Lett. 95, 210502 (2005). [5] P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, Nature 434, 169 (2005). [6] T. P. Bodiya and L.-M. Duan, Phys. Rev. Lett. 97, 143601 (2006). [7] V. Danos and E. Kashefi, Phys. Rev. A 74, 052310 (2006). [8] R. Prevedel, P. Walther, F. Tiefenbacher, P. B¨ ohi,

R. Kaltenbaek, T. Jennewein, and A. Zeilinger, Nature 445, 65 (2007). S. Popescu, Phys. Rev. Lett. 99, 250501 (2007). M. Varnava, D. E. Browne, and T. Rudolph, Phys. Rev. Lett. 100, 060502 (2008). C.-Y. Lu, X.-Q. Zhou, O. G¨ uhne, W.-B. Gao, J. Zhang, Z.-S. Yuan, A. Goebel, T. Yang, and J.-W. Pan, Nature Phys. 3, 91 (2007). A. Cabello, Phys. Rev. Lett. 95, 210401 (2005). V. Scarani, A. Acin, E. Schenck, and M. Aspelmeyer, Phys. Rev. A 71, 042325 (2005). G. Vallone, E. Pomarico, P. Mataloni, F. De Martini, and V. Berardi, Phys. Rev. Lett. 98, 180502 (2007). M. Nielsen and I. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2000).

Acknowledgments

[9] [10] [11] [12] [13] [14] [15]

13 [16] E. Knill, R. Laflamme, and G. J. Milburn, Nature 409, 46 (2001). [17] P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, Rev. Mod. Phys. 79, 135 (2007). [18] M. Barbieri, C. Cinelli, P. Mataloni, and F. De Martini, Phys. Rev. A 72, 052110 (2005). [19] M. Barbieri, F. De Martini, P. Mataloni, G. Vallone, and A. Cabello, Phys. Rev. Lett. 97, 140407 (2006). [20] G. Vallone, E. Pomarico, F. De Martini, and P. Mataloni, Phys. Rev. Lett. 100, 160502 (2008). [21] G. Vallone, E. Pomarico, F. De Martini, and P. Mataloni, Las. Phys. Lett. 5, 398 (2008). [22] K. Chen, C.-M. Li, Q. Zhang, Y.-A. Chen, A. Goebel, S. Chen, A. Mair, and J.-W. Pan, Phys. Rev. Lett. 99, 120503 (2007). [23] M. A. Nielsen, Cluster-state quantum computation (2005), [quant-ph/0504097v2]. [24] M. S. Tame, M. Paternostro, M. S. Kim, and V. Vedral, Phys. Rev. A. 72, 012319 (2005). [25] R. Raussendorf, D. E. Browne, and H. J. Briegel, Phys. Rev. A 68, 022312 (2003). [26] C. Cinelli, M. Barbieri, R. Perris, P. Mataloni, and F. De

Martini, Phys. Rev. Lett. 95, 240405 (2005). [27] D. Gottensman, Ph.D. thesis, CalTech, Pasadena (1997). [28] M. Hein, W. D¨ ur, J. Eisert, R. Raussendorf, M. V. den Nest, and H.-J. Briegel, in Quantum computers, algorithms and chaos, edited by P. Zoller, G. Casati, D. Shepelyansky, and G. Benenti (2006), International School of Physics Enrico Fermi (Varenna, Italy), [quant-ph/0602096]. [29] L. K. Grover, Phys. Rev. Lett. 79, 325 (1997). [30] L. K. Grover, Phys. Rev. Lett. 79, 4709 (1997). [31] M. S. Tame, R. Prevedel, M. Paternostro, P. Bohi, M. S. Kim, and A. Zeilinger, Phys. Rev. Lett. 98, 140501 (2007). [32] Note that the usual post-selection needed to select out the vacuum state is in any case necessary. This is unavoidable since SPDC is a not deterministic process. [33] This is a generic rotation iff the input state |χin i is not |0i or |1i. In our case the algorithm is implemented with |χin i = |±i. In fact three sequential rotations are in general necessary to implement a generic SU (2) matrix but only two, namely Rx (β)Rz (α), are sufficient to rotate the input state |χin i = |±i into a generic state